
Since \(\mathbb {E}_{\pi }\left\Vert \nabla \log \frac{p_{t}}{\pi }\right\Vert ^{2}\ge 0\) ,
the identity (REF ) exhibits that KL divergence with
respect to \(\pi \)  is diminishing along the Langevin dynamics, thus
the distribution \(p_{t}\)  actually converges to \(\pi \) . When \(\pi \) 
fulfills log-Sobolev inequality (LSI), [1]} Lemma
2 shows that
\(H(p_{t}|\pi )\le e^{-2\gamma t}H(p_{0}|\pi ).\) 
